GT0007

euclidean_gcd_step_backward

A relational gcd of the previous exact Euclidean pair is a relational gcd of the next pair.

Alpha v34 checked-use · first admitted v22 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

G101 was OPEN at this family's Alpha-v22 first admission: its complete anchored trace and actual terminal gcd were proved but its logarithmic bound was not. G101 is now CLOSED in Alpha v23, including the exact first-order bound steps≤2*BitLen(b)+1.

Exact theorem in conservative defined notation

∀ g. ∀ a. ∀ b. ∀ q. ∀ r. EuclideanDivision(a,b,q,r) → IsGCD(g,a,b) → IsGCD(g,b,r)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

EuclideanDivision(a,b,q,r) · 1IsGCD(g,a,b) · 2

Actual proof prerequisites

is_gcd_euclid_backward · checked external prerequisite
Original expanded first-order statement 
forall g a b q r. ((a = b * q + r /\ (exists ff_lt_ec_egt_step_division. ff_lt_ec_egt_step_division + S r = b))) -> ((((exists hag_left_factor_egt_before. a = g * hag_left_factor_egt_before) /\ (exists hag_right_factor_egt_before. b = g * hag_right_factor_egt_before)) /\ forall hag_divisor_egt_before. (exists hag_common_left_egt_before. a = hag_divisor_egt_before * hag_common_left_egt_before) -> (exists hag_common_right_egt_before. b = hag_divisor_egt_before * hag_common_right_egt_before) -> exists hag_greatest_factor_egt_before. g = hag_divisor_egt_before * hag_greatest_factor_egt_before)) -> ((((exists hag_left_factor_egt_after. b = g * hag_left_factor_egt_after) /\ (exists hag_right_factor_egt_after. r = g * hag_right_factor_egt_after)) /\ forall hag_divisor_egt_after. (exists hag_common_left_egt_after. b = hag_divisor_egt_after * hag_common_left_egt_after) -> (exists hag_common_right_egt_after. r = hag_divisor_egt_after * hag_common_right_egt_after) -> exists hag_greatest_factor_egt_after. g = hag_divisor_egt_after * hag_greatest_factor_egt_after))

Complete unchanged native tactic proof

All 16 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

16 script commands · 3 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–7

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro g
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro q
  5. L5
    intro r
  6. L6
    intro hstep
  7. L7
    intro hg
02Separate the logical casesL8–8

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L8
    cases hstep
03Use earlier factsL9–16

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L9
    specialize is_gcd_euclid_backward g
  2. L10
    specialize is_gcd_euclid_backward a
  3. L11
    specialize is_gcd_euclid_backward b
  4. L12
    specialize is_gcd_euclid_backward q
  5. L13
    specialize is_gcd_euclid_backward r
  6. L14
    apply is_gcd_euclid_backward
  7. L15
    exact hstep_left
  8. L16
    exact hg

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro g
  2. 0002intro a
  3. 0003intro b
  4. 0004intro q
  5. 0005intro r
  6. 0006intro hstep
  7. 0007intro hg
  8. 0008cases hstep
  9. 0009specialize is_gcd_euclid_backward g
  10. 0010specialize is_gcd_euclid_backward a
  11. 0011specialize is_gcd_euclid_backward b
  12. 0012specialize is_gcd_euclid_backward q
  13. 0013specialize is_gcd_euclid_backward r
  14. 0014apply is_gcd_euclid_backward
  15. 0015exact hstep_left
  16. 0016exact hg